## SectionB.1Introduction

Set theory is concerned with elements, certain collections of elements called sets and a concept of membership. For each element $$x$$ and each set $$X\text{,}$$ exactly one of the following two statements holds:

1. $$x$$ is a member of $$X\text{.}$$

2. $$x$$ is not a member of $$X\text{.}$$

It is important to note that membership cannot be ambiguous.

When $$x$$ is an element and $$X$$ is a set, we write $$x\in X$$ when $$x$$ is a member of $$X\text{.}$$ Also, the statement $$x$$ belongs to $$X$$ means exactly the same thing as $$x$$ is a member of $$X\text{.}$$ Similarly, when $$x$$ is not a member of $$X\text{,}$$ we write $$x\notin X$$ and say $$x$$ does not belong to $$X\text{.}$$

Certain sets will be defined explicitly by listing the elements. For example, let $$X=\{a,b,d,g,m\}\text{.}$$ Then $$b\in X$$ and $$h\notin X\text{.}$$ The order of elements in such a listing is irrelevant, so we could also write $$X=\{g,d,b,m,a\}\text{.}$$ In other situations, sets will be defined by giving a rule for membership. As examples, let $$\posints$$ denote the set of positive integers. Then let $$X=\{n\in\posints:5\le n\le 9\}\text{.}$$ Note that $$6,8\in X$$ while $$4,10,238\notin X\text{.}$$

Given an element $$x$$ and a set $$X\text{,}$$ it may at times be tedious and perhaps very difficult to determine which of the statements $$x\in X$$ and $$x\notin X$$ holds. But if we are discussing sets, it must be the case that exactly one is true.

### ExampleB.1.

Let $$X$$ be the set consisting of the following $$12$$ positive integers:

\begin{align*} \amp 13232112332\\ \amp 13332112332\\ \amp 13231112132\\ \amp 13331112132\\ \amp 13232112112\\ \amp 13231112212\\ \amp 13331112212\\ \amp 13232112331\\ \amp 13231112131\\ \amp 13331112131\\ \amp 13331112132\\ \amp 13332112111\\ \amp 13231112131 \end{align*}

Note that one number is listed twice. Which one is it? Also, does $$13232112132$$ belong to $$X\text{?}$$ Note that the apparent difficulty of answering these questions stems from (1) the size of the set $$X$$ and (2) the size of the integers that belong to $$X\text{.}$$ Can you think of circumstances in which it is difficult to answer whether $$x$$ is a member of $$X$$ even when it is known that $$X$$ contains exactly one element?

### ExampleB.2.

Let $$P$$ denote the set of primes. Then $$35\notin P$$ since $$35= 5\times 7\text{.}$$ Also, $$19\in P\text{.}$$ Now consider the number

\begin{equation*} n = 77788467064627123923601532364763319082817131766346039653933 \end{equation*}

Does $$n$$ belong to $$P\text{?}$$ Alice says yes while Bob says no. How could Alice justify her affirmative answer? How could Bob justify his negative stance? In this specific case, I know that Alice is right. Can you explain why?